Q 5 - Calculus

Question

What are the graphs of the constant functions$\rho ={\rho }_0$

$\theta ={\theta }_0$, and $\phi ={\phi }_0$ in the spherical coordinate system?

Step-by-Step Solution

Answer

The required graph is shown:

The equation$\theta ={\theta }_0$ describes a vertical half-plane with the $z-$ axis as its origin, and $\varphi ={\varphi }_0$represent a cone with the z-axis, and as symmetry $\overrightarrow{OP}$ of the $z-$axis. 

Step 1: Given information

The ordered triple represents points in spherical coordinates and is denoted by.$(\rho ,\theta ,\varphi )$

Here,$\rho$ is the distance between the point and the origin$O$, $\theta$where is the angle in radians between the positive $x$-axis and the line segment$xy-$ connecting the origin and the point, and$\varphi$ is the angle between the positive -$z$axis and the line segment$\overrightarrow{OP}$ connecting the origin and the point$P(\rho ,\theta ,\varphi )$

Step 2: Draw a graph with the help of given equations

The graph is displayed below.

The equation$\theta ={\theta }_0$ describes a vertical half-plane with the -$z$ axis as its origin, and a cone with the z-axis as its symmetry$\overrightarrow{OP}$of $z-$ axis

Step 3: Determine whether the given functions is represented a coordinate system or not

A sphere centred at the origin is represented in the spherical coordinates system$\rho ={\rho }_0$

The equation$\theta ={\theta }_0$ describes a vertical half-plane with the $z$- axis as its origin, and$\varphi ={\varphi }_0$represent a cone with the z-axis and as symmetry $\overrightarrow{OP}$ of $z$-axis.